274 On the Mathematical Theory of Oceanic Tides. 



Tahiti and the Sandwich Islands " (p. 6) ; and that " the tides 

 over a great portion of the Pacific are so small that we may con- 

 sider the lunar tide as almost vanishing " (p. 18). It is evident 

 that the conditions supposed in the theory are likely to be more 

 approximately fulfilled at the central portion of this large ocean 

 than at any other part of the earth's surface. 



Further, it may be remarked that the principal term in the 

 expression for r, which varies as cos 2 X, is constant for a given 

 latitude only because the moon was supposed to be in the equa- 

 tor. If the problem were treated more generally, and the moon's 

 declination were taken into account, it may be presumed that 

 there would be a corresponding term varying with the declina- 

 tion ; and as the variation would be continuous, and the term 

 would represent the principal amount of tidal elevation, the 

 effect might be exhibited as a sensible diurnal variation of the 

 tide. Whewell has, in fact, made mention, in the memoir above 

 cited, of " a diurnal inequality following the changes of the 

 moon's declination." 



At the end of my communication on this problem in the Ja- 

 nuary Number I said (as it now appears, on insufficient grounds) 

 that the solution it contained was " strictly based on the neces- 

 sary principles of hydrodynamics." My object in both attempts 

 has been to point out a course of reasoning by which this great 

 desideratum in the theoretical treatment of tides might be at- 

 tained. The present mathematical theory, by clearing up a dif- 

 ficulty relating to the logical method of applying the general 

 equation (a), has, I think, materially contributed towards ac- 

 complishing this object. The difficulty I refer to consisted in 

 obtaining impossible results by applying that equation in a 

 manner accordant with received hydrodynamical principles. As 

 the proposed explanation of it introduces a new and important 

 principle in the application of analysis to hydrodynamics, I 

 have thought it worth while to call attention to the reasoning 

 employed by stating it here again as succinctly as possible. 



Since the general equation (a), or <E> = 0, is applicable to all 

 parts of the fluid at all times, it would still be true if to given 

 values of x, y, z, and t the small independent variations 8x, hy, 

 &z, and St were added. That is, we shall have O + 84> = 0, as 

 well as <£> = (), for the same values of x, y, z } and t. Hence we 

 have a new general equation 8<I>=0. The general integral of 

 <3>=0, obtained independently of any particular case of motion, 

 must satisfy both <I> = and S<I> = 0. But a particular solution 

 which applies to the circumstances of a given case of motion is 

 only required to satisfy £<l> = because it coincides with the 

 general integral only through an indefinitely small space for an 

 indefinitely small time; and if it also satisfied the equation <I>=0, 



